Monte Carlo methods for option pricing

Overview

In mathematical finance, a Monte Carlo option model uses Monte Carlo methods to calculate the value of an option with multiple sources of uncertainty or with complicated features. The first application to option pricing was by Phelim Boyle in 1977 (for European options). In 1996, M. Broadie and P. Glasserman showed how to price Asian options by Monte Carlo. An important development was the introduction in 1996 by Carriere of Monte Carlo methods for options with early exercise features.

Methodology

As is standard, Monte Carlo valuation relies on risk neutral valuation. Here the price of the option is its discounted expected value; see risk neutrality and rational pricing. The technique applied then, is (1) to generate a large number of possible, but random, price paths for the underlying (or underlyings) via simulation, and (2) to then calculate the associated exercise value (i.e. "payoff") of the option for each path. (3) These payoffs are then averaged and (4) discounted to today. This result is the value of the option. This approach, although relatively straightforward, allows for increasing complexity:

An option on equity may be modelled with one source of uncertainty: the price of the underlying stock in question. Here the price of the underlying instrument

S

t

{\displaystyle \ S_{t}\,}

is usually modelled such that it follows a geometric Brownian motion with constant drift

μ

{\displaystyle \mu \,}

and volatility

σ

{\displaystyle \sigma \,}

. So:

d

S

t

= μ

S

t

d t + σ

S

Least Square Monte Carlo

Least Square Monte Carlo is a technique for valuing early-exercise options (i.e. Bermudan or American options). It was first introduced by Jacques Carriere in 1996. It is based on the iteration of a two step procedure:

First, a backward induction process is performed in which a value is recursively assigned to every state at every timestep. The value is defined as the least squares regression against market price of the option value at that state and time (-step). Option value for this regression is defined as the value of exercise possibilities (dependent on market price) plus the value of the timestep value which that exercise would result in (defined in the previous step of the process). Secondly, when all states are valued for every timestep, the value of the option is calculated by moving through the timesteps and states by making an optimal decision on option exercise at every step on the hand of a price path and the value of the state that would result in. This second step can be done with multiple price paths to add a stochastic effect to the procedure.

Application

As can be seen, Monte Carlo Methods are particularly useful in the valuation of options with multiple sources of uncertainty or with complicated features, which would make them difficult to value through a straightforward Black–Scholes-style or lattice based computation. The technique is thus widely used in valuing path dependent structures like lookback- and Asian options and in real options analysis. Additionally, as above, the modeller is not limited as to the probability distribution assumed. Conversely, however, if an analytical technique for valuing the option exists—or even a numeric technique, such as a (modified) pricing tree—Monte Carlo methods will usually be too slow to be competitive. They are, in a sense, a method of last resort; see further under Monte Carlo methods in finance. With faster computing capability this computational constraint is less of a concern.

See also

Monte Carlo methods in finance Quasi-Monte Carlo methods in finance Stochastic modelling (insurance) Stochastic asset model

References

Notes

Sources

Primary references

Boyle, Phelim P. (1977). "Options: A Monte Carlo Approach". Journal of Financial Economics. 4 (3): 323–338. doi:10.1016/0304-405x(77)90005-8. Retrieved June 28, 2012. Broadie, M.; Glasserman, P. (1996). "Estimating Security Price Derivatives Using Simulation" (PDF). Management Science. 42 (2): 269–285. CiteSeerX 10.1.1.196.1128. doi:10.1287/mnsc.42.2.269. Retrieved June 28, 2012. {{cite journal}}: Cite uses deprecated parameter |citeseerx= (help) Longstaff, Francis A.; Schwartz, Eduardo S. (2001). "Valuing American options by simulation: a simple least squares approach". Review of Financial Studies. 14: 113–148. CiteSeerX 10.1.1.155.3462. doi:10.1093/rfs/14.1.113. Retrieved June 28, 2012. {{cite journal}}: Cite uses deprecated parameter |citeseerx= (help) Bibliography

Bruno Dupire (1998). Monte Carlo:methodologies and applications for pricing and risk management. Risk. Paul Glasserman (2003). Monte Carlo methods in financial engineering. Springer-Verlag. ISBN 978-0-387-00451-8. Peter Jaeckel (2002). Monte Carlo methods in finance. John Wiley and Sons. ISBN 978-0-471-49741-7. Don L. McLeish (2005). Monte Carlo Simulation & Finance. ISBN 978-0-471-67778-9. Christian P. Robert, George Casella (2004). Monte Carlo Statistical Methods. ISBN 978-0-387-21239-5.

External links

Online tools

Monte Carlo simulated stock price time series and random number generator (allows for choice of distribution), Steven Whitney Discussion papers and documents

Monte Carlo Simulation, Prof. Don M. Chance, Louisiana State University Pricing complex options using a simple Monte Carlo Simulation, Peter Fink (reprint at quantnotes.com) MonteCarlo Simulation in Finance, global-derivatives.com Monte Carlo Derivative valuation, contd., Timothy L. Krehbiel, Oklahoma State University–Stillwater Applications of Monte Carlo Methods in Finance: Option Pricing, Y. Lai and J. Spanier, Claremont Graduate University Option pricing by simulation, Bernt Arne Ødegaard, Norwegian School of Management Pricing and Hedging Exotic Options with Monte Carlo Simulations, Augusto Perilla, Diana Oancea, Prof. Michael Rockinger, HEC Lausanne